Any stopping time with expected alarm time at least b is the level-b crossing of a weak e-detector, and any stopping time satisfying optional-horizon false-alarm control is the level-b crossing of a strong e-detector.
No monotonicity assumption is needed for the representation theorem, although in applicationsw will often be nonincreasing so that early alarms are penalized more heavily
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Universality of e-detectors for ARL control
Any stopping time with expected alarm time at least b is the level-b crossing of a weak e-detector, and any stopping time satisfying optional-horizon false-alarm control is the level-b crossing of a strong e-detector.